Chaos - James Gleick [35]
At first, May could not see this whole picture. But the fragments he could calculate were unsettling enough. In a real-world system, an observer would see just the vertical slice corresponding to one parameter at a time. He would see only one kind of behavior—possibly a steady state, possibly a seven-year cycle, possibly apparent randomness. He would have no way of knowing that the same system, with some slight change in some parameter, could display patterns of a completely different kind.
James Yorke analyzed this behavior with mathematical rigor in his “Period Three Implies Chaos” paper. He proved that in any one-dimensional system, if a regular cycle of period three ever appears, then the same system will also display regular cycles of every other length, as well as completely chaotic cycles. This was the discovery that came as an “electric shock” to physicists like Freeman Dyson. It was so contrary to intuition. You would think it would be trivial to set up a system that would repeat itself in a period-three oscillation without ever producing chaos. Yorke showed that it was impossible.
Startling though it was, Yorke believed that the public relations value of his paper outweighed the mathematical substance. That was partly true. A few years later, attending an international conference in East Berlin, he took some time out for sightseeing and went for a boat ride on the Spree. Suddenly he was approached by a Russian trying urgently to communicate something. With the help of a Polish friend, Yorke finally understood that the Russian was claiming to have proved the same result. The Russian refused to give details, saying only that he would send his paper. Four months later it arrived. A. N. Sarkovskii had indeed been there first, in a paper titled “Coexistence of Cycles of a Continuous Map of a Line into Itself.” But Yorke had offered more than a mathematical result. He had sent a message to physicists: Chaos is ubiquitous; it is stable; it is structured. He also gave reason to believe that complicated systems, traditionally modeled by hard continuous differential equations, could be understood in terms of easy discrete maps.
WINDOWS OF ORDER INSIDE CHAOS. Even with the simplest equation, the region of chaos in a bifurcation diagram proves to have an intricate structure—far more orderly than Robert May could guess at first. First, the bifurcations produce periods of 2, 4, 8, 16…. Then chaos begins, with no regular periods. But then, as the system is driven harder, windows appear with odd periods. A stable period 3 appears, and then the period-doubling begins again 6, 12, 24…. The structure is infinitely deep. When portions are magnified, they turn out to resemble the whole diagram.
The sightseeing encounter between these frustrated, gesticulating mathematicians was a symptom of a continuing communications gap between Soviet and Western science. Partly because of language, partly because of restricted travel on the Soviet side, sophisticated Western scientists have often repeated work that already existed in the Soviet literature. The blossoming of chaos in the United States and Europe has inspired a huge body of parallel work in the Soviet Union; on the other hand, it also inspired considerable bewilderment, because much of the new science was not so new in Moscow. Soviet mathematicians and physicists had a strong tradition in chaos research, dating back to the work of A. N. Kolmogorov in the fifties. Furthermore, they had a tradition of working together that had survived the divergence of mathematics and physics elsewhere.
Thus Soviet scientists were receptive to Smale—his horseshoe created a considerable stir in the sixties. A brilliant mathematical physicist, Yasha Sinai, quickly translated similar systems into thermodynamic terms. Similarly, when Lorenz’s work finally reached Western physics